The system of differential equations given by dx dt dy dt has the general solution [0]=0₁1] α1 -3t = -4x+6y = -x + 3y + a2 He²t, a1, a2 € R. (a) Sketch a phase portrait of the system near the critical point at the origin. You should include the following in your sketch, and explain your reasoning to justify your conclusions: any straight line orbits and their directions; at least 4 other orbits and their directions, showing the asymptotic behaviour as t→→ ∞ and t→-∞; • the slopes at which the orbits cross the x and y axes.
The system of differential equations given by dx dt dy dt has the general solution [0]=0₁1] α1 -3t = -4x+6y = -x + 3y + a2 He²t, a1, a2 € R. (a) Sketch a phase portrait of the system near the critical point at the origin. You should include the following in your sketch, and explain your reasoning to justify your conclusions: any straight line orbits and their directions; at least 4 other orbits and their directions, showing the asymptotic behaviour as t→→ ∞ and t→-∞; • the slopes at which the orbits cross the x and y axes.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:The system of differential equations given by
dx
dt
dy
dt
has the general solution
[x(t)
= α1
8
-3t
e
= - 4x + 6y
= -x + 3y
He²t.
(a) Sketch a phase portrait of the system near the critical point at the origin.
You should include the following in your sketch, and explain your reasoning to justify your
conclusions:
+02
a1, a2 € R.
• any straight line orbits and their directions;
• at least 4 other orbits and their directions, showing the asymptotic behaviour as
t→→ ∞ and t→→∞;
the slopes at which the orbits cross the x and y axes.
(b) What is the type and stability of the critical point for this system?
(c) If y(0) = 0 and y(1) = 1, compute y(2).
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